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Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

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Abstract

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials \({P_{L,n}(x), \ x\in[0,L]}\) is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre–Gauss–Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge–Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    A. H. Bhrawy & M. A. Abdelkawy

  2. Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, El-Kharja, 72511, Egypt

    S. S. Ezz-Eldien

Authors
  1. A. H. Bhrawy
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  2. M. A. Abdelkawy
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  3. S. S. Ezz-Eldien
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Correspondence to A. H. Bhrawy.

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Bhrawy, A.H., Abdelkawy, M.A. & Ezz-Eldien, S.S. Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions. Mediterr. J. Math. 13, 2483–2506 (2016). https://doi.org/10.1007/s00009-015-0635-y

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  • DOI: https://doi.org/10.1007/s00009-015-0635-y

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